Optimal Transportation Problems and a Monge-Ampere Type Equation

Abstract

In this thesis, we study the theory of optimal transportation problems. First, we prepare basic theories on optimal transportation problems, including the formulation of the Monge problem, Kantorovich problem, and the dual problem. Besides, we also demonstrate the existence of a solution and its uniqueness under certain conditions. Next, in the second chapter, we establish the connection between optimal transportation and a particular type of Monge-Ampere equation, based on the paper by Ma, Trudinger and Wang [10]. Finally, in the third chapter, we present a proof due to [6] and [10] on regularity of the solution of this Monge-Ampere type equation under some special conditions on the cost function. The thesis can serve as a guide for any interested readers in the paper [10], where the famous Ma-Trudinger-Wang condition on cost function was first proposed. Show more